Generation of semigroup for symmetric matrix Schrödinger operators in $L^p$-spaces

Abstract

In this paper we establish generation of analytic strongly continuous semigroup in $L^p$--spaces for the symmetric matrix Schrödinger operator $div(Q\nabla u)-Vu$, where, for every $x\in {\mathbb{R}}^d$, $V(x)=(v_{ij}(x))$ is a semi-definite positive and symmetric matrix. The diffusion matrix $Q(\cdot )$ is supposed to be strongly elliptic and bounded and the potential $V$ satisfies the weak condition $v_{ij}\in L_{loc}^1({\mathbb{R}}^d)$, for all $i,j\in \{1,\dots ,m\}$. We also characterize positivity of the semigroup and we investigate on its compactness.

11 pages, no figures

References in corpus (1)

• Superadiabatic transition histories in quantum molecular dynamics